Quantum chaos for point scatterers on flat tori

Ueberschär, Henrik

doi:10.1098/rsta.2012.0509

Cited by 15 publications

(21 citation statements)

References 29 publications

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“…Notably, this forces a re-normalization of (1.1) tan(ϕ/2) m≥1 r(m) m 2 + 1 ∼ −π log λ so that ϕ depends on λ in this case (see [47] equation (3.14)). We note that the weak coupling quantization corresponds to a fixed self adjoint extension, whereas the strong coupling quantization can be viewed as an energy dependent, albeit very slowly varying, family of self adjoint extensions.…”

Section: Introductionmentioning

confidence: 99%

Superscars for Arithmetic Toral Point Scatterers

Kurlberg

Rosenzweig

2016

Commun. Math. Phys.

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We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus T 2 = R 2 /Z 2 . Given any probability measure arising by placing delta masses, with equal weights, on Z 2 -lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and are completely delocalized in position (i.e., concentration on Lagrangian states.) We also show that the mass, in momentum, can fully localize on more exotic measures, e.g. singular continous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals.Date: December 16, 2019. 1 probability measures on the unit circle, is of the form cµ sing + (1 − c)µ uniform , for some c ∈ [1/2, 1]. Here both µ uniform and µ sing are normalized to have mass one, and µ sing can be taken to be a sum of delta measures giving equal mass to the four points ±(1, 0), ±(0, 1). We note that µ uniform is the push-forward of the Liouville measure and hence maximally delocalized, whereas µ sing is maximally localized since any quantum limits in this setting must be invariant under a certain eight fold symmetry (cf. (1.5)).Stronger localization, i.e., going strictly beyond c = 1/2, is particularly interesting given a number of "half delocalization" results for quantum limits for some other (strongly chaotic) systems, namely quantized cat maps and geodesic flows on manifolds with constant negative curvature. For example, in the former case Faure and Nonnenmacher showed [12] that if a quantum limit ν is decomposed as ν = ν pp + ν Liouville + ν sc , with ν pp denoting the pure point part and ν sc denoting the singular continous part, then ν Liouville (T 2 ) ≥ ν pp (T 2 ), and thus ν pp (T 2 ) ≤ 1/2. (We emphasize that T 2 is the full phase space in this setting.)The aim of this paper is to exhibit strong, essentially maximal, localization for a quantum ergodic system, namely arithmetic toral point scatterers. In particular we construct quantum limits (in momentum) corresponding to c = 1 in the above decomposition; other interesting examples include singular continous measures with support, say, on Cantor sets. This can be viewed as a step towards a "measure classification" for quantum limits of quantum ergodic systems.1.1. Description of the model. Let us now describe the basic properties of the point scatterer. This is discussed in further detail in [37,38,28,26,39,41]. To describe the quantum system associated with the point scatterer, consider −∆| Dx 0 where 5

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Section: Introductionmentioning

confidence: 99%

Superscars for Arithmetic Toral Point Scatterers

Kurlberg

Rosenzweig

2016

Commun. Math. Phys.

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“…Note that the formulation of Theorem 1.3 is fairly general, and is independent of the self-adjoint extension U , which is advantageous since in the physics literature one often considers self-adjoint extensions which are not fixed but vary with λ. For a single scatterer, for example, there is a popular quantization condition known as the "strong coupling limit" where tan φ 2 ∼ −C log λ (see [11,13]), in which phenomena such as level repulsion between the new eigenvalues are observed, as opposed to the "weak coupling limit" where the self-adjoint extension is fixed. In particular, it follows from Theorem 1.3 that uniform distribution in configuration space holds even if the self-adjoint extensions change with λ.…”

Section: Statement Of the Main Resultmentioning

confidence: 99%

Uniform distribution of eigenstates on a torus with two point scatterers

Yesha

2018

J. Spectr. Theory

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We study the Laplacian perturbed by two delta potentials on a two-dimensional flat torus. There are two types of eigenfunctions for this operator: old, or unperturbed eigenfunctions which are eigenfunctions of the standard Laplacian, and new, perturbed eigenfunctions which are affected by the scatterers. We prove that along a density one sequence, the new eigenfunctions are uniformly distributed in configuration space, provided that the difference of the scattering points is Diophantine.

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“…Such an extension is not unique and we can parametrize all possible self-adjoint extensions with a single parameter α ∈ (−∞, ∞] which can be considered as a number related to the strength of the point scatterer. See [3,4,5,6] for further developments. In this paper we consider a point scatterer on a thin rectangle.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of Šeba billiards with localization of low-energy eigenfunctions

Lee¹

2016

J. Phys. A: Math. Theor.

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We study the localization of eigenfunctions produced by a point scatterer on a thin rectangle. We find an explicit set of eigenfunctions localized to part of the rectangle by showing that the one-dimensional Schrödinger operator with a Dirac delta potential asymptotically governs the spectral properties of the two-dimensional point scatterer. We also find the rate of localization in terms of the aspect ratio of the rectangle. In addition, we present numerical results regarding the asymptotic behavior of the localization.

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Quantum chaos for point scatterers on flat tori

Cited by 15 publications

References 29 publications

Superscars for Arithmetic Toral Point Scatterers

Superscars for Arithmetic Toral Point Scatterers

Uniform distribution of eigenstates on a torus with two point scatterers

Eigenvalues of Šeba billiards with localization of low-energy eigenfunctions

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